w_{\infty} and sl_q(2) Algebras in the Landau Problem and Chern-Simons Theory on a Torus

TL;DR

This paper explores the emergence of multiple $w_ Infty$ and $sl_q(2)$ algebras in Chern-Simons theory and the Landau problem on a torus when the flux or Chern-Simons coefficient is rational, revealing new algebraic structures.

Contribution

It demonstrates the existence of two sets of $w_ Infty$ and $sl_q(2)$ algebras under rational flux or Chern-Simons coefficient, extending previous single-set results.

Findings

Multiple $w_ Infty$ and $sl_q(2)$ algebras appear for rational flux or Chern-Simons coefficient.

General wavefunctions for the Landau problem with rational flux are constructed.

The algebraic structure depends on the rationality of the flux or coefficient.

Abstract

We discuss $w_\infty$ and $sl_q(2)$ symmetries in Chern-Simons theory and Landau problem on a torus. It is shown that when the coefficient of the Chern-Simons term, or when the total flux passing through the torus is a rational number, there exist in general two $w_\infty$ and two $sl_q(2)$ algebras, instead of one set each discussed in the literature. The general wavefunctions for the Landau problem with rational total flux is also presented.